Objective.In ultrasound time-harmonic elastography (THE), tissue elasticity is estimated from the ultrasound-measured shear-wave displacement fields, generated by an external vibrator. Commonly used techniques to map from the ultrasound pulse-echo data to tissue elasticity maps, such as the phase-gradient and local-frequency estimation methods, base their estimates on a set of displacement field data and do not involve direct numerical modeling of the underlying shear-wave propagation. In the current proof-of-concept study, the objective is to develop a full-waveform inversion (FWI) approach to ultrasound-based THE, providing a physics-driven framework for improved tissue elasticity estimation.Approach.FWI is a methodology commonly applied in geophysics. It relies on simulating the wave propagation and scattering in heterogeneous media to model the intricate relationship between the observed wavefield and the propagation medium properties. Our FWI-based tissue-harmonic elastography approach is based on minimizing a cross-correlation based objective function that quantifies phase mismatches between measured and simulated shear waves.MainResults.Experiments on synthetic phantom data obtained from the fractional Kelvin-Voigt wave model demonstrate that our approach accurately reconstructs the shear wave speed of the medium, even in the presence of significant noise, and without relying on pre-filtering steps such as directional filtering, which selects waves based on their propagation direction. An improvement of approximately 0.4 m⋅s-1within the inclusion region, compared to an established method, is observed for a noisy wavefield with decreasing signal-to-noise ratio, ranging from 20 to-10dB from top to bottom of the ultrasound image. In addition, the reconstructed shear-wave speed map achieved a 50% reduction in standard deviation.Significance.This work demonstrates the potential for enhanced elasticity mapping using FWI-based reconstruction in THE with prospective impacts for future clinical applications.

Wavenumber-domain beamforming offers substantial computational advantage over conventional time-domain algorithms in ultrasound image formation. However, these methods are found to introduce distortions at large angles and greater depths, making it nontrivial to replace conventional processing pipelines with wavenumber-domain approaches. This paper presents a wavenumber-domain framework that produces images consistent with conventional delay-and-sum (DAS) beamforming, while retaining the computational efficiency of Fourier-domain processing. Building on recent work on DAS-consistent imaging for multistatic acquisition data, the key utility of this novel algorithm is its applicability in plane-wave, focused-transmit, and diverging-wave imaging. The approach compensates for the implicit spatial-frequency filtering of earlier wavenumber-domain formulations through a modified Fourier-domain weighting and an axial scaling to the reconstructed image. Using data from simulations, phantom experiments, and in-vivo liver imaging, we demonstrate that the proposed method preserves DAS-equivalent amplitude, speckle statistics, resolution, and contrast. Quantitative image comparison confirms a close agreement between DAS and the proposed method with Structural Similarity (SSIM) values exceeding 0.99, whereas SSIM between earlier wavenumber-domain beamformers and DAS is lower, ranging from 0.67 to 0.92 depending on the transmit scheme. The proposed method yields approximately one order-of-magnitude reduction in computational cost compared to conventional DAS, making it well-suited for real-time and resource-constrained ultrasound systems.

Window functions are commonly used to balance the mainlobe width and sidelobe levels in beamforming applications. Traditionally, windows are selected independently for the transmit (Tx) and receive (Rx) side. However, in active systems like radar, medical ultrasound, and sonar, the same array may operate in both modes, and the convolution of the Tx and Rx windows determines the effective two-way response. Therefore, without jointly designing the Tx/Rx windows, the mainlobe and sidelobe characteristics do not fully benefit from the possibilities of a consolidated two-way approach. This paper presents a framework for jointly designing Tx/Rx window functions by factorization of any desired two-way effective aperture into separate Tx/Rx windows. To guide this factorization, we introduce the white noise gain product (WNGP), a metric quantifying the combined spatial filtering efficiency of the Tx/Rx pair. We then propose a root-allocation strategy for the factorization that maximizes this metric, enabling effective control over the two-way beampattern. Additionally, we derive a recursive formula for the optimal Tx/Rx windows that yield a uniform effective aperture. The approach is validated through simulations, showing improved sidelobe control and greater design flexibility compared to conventional windowing techniques.

Software beamforming has enabled the introduction of a myriad adaptive beamformers. To reproduce and compare those techniques, a unified beamforming framework is imperative. Here, we propose such a framework, called the Generalized Beamformer (GB), which provides a unified approach to most state-of-the-art beamforming techniques through a single core expression. This generalization is achieved through a range of delay and apodization models that enable translation across various transmit sequences. Exploiting the GB we also demonstrate a novel synthetic transmit focusing strategy for adaptive beamforming, where signals are coherently combined across transmit events prior to e.g. Coherence Factor or Minimum Variance processing. For the investigated CPWC case, this substantially reduces computational complexity while improving resolution and maintaining contrast. The GB is implemented within the open-source UltraSound ToolBox (USTB) for MATLAB, and its pixel-based approach provides flexibility in scan grid definition. The framework is shown to successfully beamform conventional transmit sequences, including focused, diverging, plane, and single-element transmissions. Its versatility is showcased through diverse applications, including in-vivo cardiac and fetal imaging, as well as subsea sonar.

Fourier-domain beamforming methods offer a computationally efficient alternative to time domain methods in ultrasound imaging. Among these, the wavenumber algorithm (WA) achieves fast image reconstruction from full-matrix capture (FMC) data but introduces amplitude distortions due to its inherent weighting. In this study, we present a modified Fourier-domain beamforming method that operates on FMC data and reproduces the amplitude response of the conventional delay-and-sum (DAS) method while maintaining the computational efficiency of WA. This is achieved through a derivation of a correction factor that restores DAS-like amplitude characteristics in WA. The correction is implemented in two stages, consisting of a matrix multiplication to the data in the Fourier domain and a subsequent scaling with the z coordinate in the reconstructed image. Validation on simulated point targets, phantom experiment, and in-vivo imaging demonstrates that the proposed method produces images nearly indistinguishable from DAS but with significantly reduced computational burden. We also investigate the role of zero-padding, showing how interpolation accuracy affects reconstruction fidelity and computational efficiency. The proposed approach can, for example, facilitate the development of ultrasound systems with reduced power consumption.

Ultrasound time-harmonic elastography (THE) estimates tissue mechanical properties by imaging continuously driven shear waves. Systematic comparison and validation of ultrasound THE methods are limited by the lack of controllable ultrasound data with known shear-wave motion and material properties. In this work, we present a simulation framework for generating synthetic ultrasound RF channel data for THE. Numerically simulated shear-wave motion is converted into time-varying scatterer distributions, which are used in pulse-echo ultrasound simulations to produce speckle-consistent data. This moving-scatterer representation provides an explicit link between shear-wave propagation modeling and ultrasound image formation while retaining access to the underlying ground-truth motion field. The framework is demonstrated using a phantom-mimicking geometry and a realistic three-dimensional liver model. The generated channel data are processed using conventional beamforming, autocorrelation-based particle-velocity estimation and k-MDEV shear-wave speed reconstruction to illustrate their use in downstream THE analysis. The examples illustrate experimentally relevant effects associated with imaging orientation, boundary reflections and ultrasound attenuation. The proposed framework provides a controlled and reproducible source of synthetic ultrasound data for developing and evaluating ultrasound acquisition, beamforming and motion-estimation methods in THE.

Wide-beam or single-transmit acquisitions often reduce local spatial coherence, breaking the narrowband model assumed by high-resolution array spectral estimators such as IAA and Capon and thereby degrading performance. We propose a discrete prolate spheroidal sequence (DPSS) subspace projection of delay-focused aperture data. This projection suppresses incoherent off-angle energy and restores local spatial coherence, enabling scanline-wise adaptive spectral estimation under severe model mismatch. Each delay-focused aperture vector is projected onto a DPSS subspace spanned by the first K eigenvectors corresponding to a small angular bandwidth. The approach is lightweight, with precomputation and a per-point complexity of O(MK) , and integrates naturally into standard delay-focused processing pipelines. Frequency-angle plots reveal how the projection reconstructs coherent ridge structures that are otherwise obscured by wide-beam incoherence. Simulations in both plane-wave and diverging-wave ultrasound scenarios demonstrate improved resolution and contrast in single-transmit wide-beam imaging. Qualitative results on recorded channel data from the public PICMUS dataset provide an experimental sanity check and validation, indicating that the same coherence-restoration behavior is observed in real recordings. All experimental validation in this work is confined to ultrasound imaging; assessment of other array-processing applications is left for future work.

In adaptive beamforming, the array signal processing adjusts its sensor delays and weights based on the incoming data. In conventional beamforming, these parameters are instead given from a predefined model. Adaptive beamformers can improve measurement precision by dynamically rejecting spatial interference. While an established theory is available on the behavior of adaptive beamformers in textbook scenarios, their expected performance on realistic pulse-echo imaging scenes is still mostly uncharted. Imaging performance can be evaluated by individual pixel precision and aggregated metrics such as resolution and contrast. The achievable gain is strongly related to the sparsity of the scene and the availability of data to appropriately estimate the spatial covariance matrix. In pulse-echo measurements, the nonstationary interference poses a special problem for adaptive beamforming, which is a current research question of academic and industrial interest. The current work establishes a performance bound for adaptive beamforming in simulated realistic pulse-echo scenarios. This is derived and numerically implemented as the clairvoyant minimum variance distortionless response beamformer. The proposed framework allows for an a priori assessment of the applicability of adaptive beamforming, for a given scenario. The performance of the implemented algorithms can be directly compared with the theoretical limit in a simulated environment.

A wide variety of transmit sequences can be employed in medical ultrasound, including plane waves, diverging waves, and focused beams. The choice of sequence often involves trade-offs between resolution, signal-to-noise ratio (SNR), frame rate, and harmonic imaging capabilities. However, the desirable mathematical property of orthogonality (i.e., absence of cross-talk) between transmits has generally received less attention. This property, often lacking, becomes particularly relevant for the recent REFoCUS (retrospective encoding for conventional ultrasound sequences) technique, which we in this work connect to the array signal processing technique called beamspace processing. Given an arbitrary transmit sequence, REFoCUS enables the recovery of signals from single-element transmissions (known as the multistatic dataset) thereby enhancing beamforming flexibility. In this context, the choice of transmit sequence influences the recovery process when using the intuitively appealing and computationally efficient adjoint-based method, which must be replaced by a regularized pseudoinverse for general applicability. In the current work, we derive the “closest” alternative to any chosen transmit sequence that makes the regularized and adjoint methods yield equal estimates of the multistatic dataset, and show via numerical experiments a reduction in beam and/or element cross-talk. The derivation is based on a matrix nearness problem of finding the nearest orthogonal (or unitary) matrix to the encoding matrix using singular value decomposition (SVD). The resulting transmit sequences offer a time-domain equivalent understanding of the regularized REFoCUS method, as well as a solution for optimizing the invertibility of ultrasound sequences.

Null subtraction imaging (NSI) is a non-linear beamformer that aims to improve the spatial resolution of ultrasound images. NSI incoherently combines three delay-and-sum (DAS) outputs from the same RF data using three related apodizations on receive. NSI has been advocated to have many advantages in different domains such as B-mode imaging, plane wave imaging, power Doppler imaging, and for large-pitch arrays. However, despite its increasing popularity, an explicit relationship between NSI resolution (interpreted as the mainlobe width) and various parameters (such as the DC offset value c, array aperture, and wavelength) is not known, making system design and intuitive reasoning about the method difficult. Therefore, in the current work, we derive the theoretical NSI array pattern and give an approximate expression for the -6dB mainlobe width. Our derivation is based on a Taylor series-expansion of the analytical NSI array pattern, which is valid over the mainlobe region for the range of c values typically seen in the literature. The results show that the NSI mainlobe width is proportional to c lambda/D , which is the DC offset value multiplied by the wavelength and divided by the aperture size, and therefore has a similar wavelengh and aperture dependency as the classical DAS mainlobe. The work is validated numerically, also showing that the NSI mainlobe width approaches the DAS mainlobe width as c approaches infinity.

Estimating the seafloor slope from interferometric synthetic aperture sonar (SAS) measurements is useful for many applications. The fundamental approach to estimating the slope in a gridded depth map is straightforward, but its statistical accuracy is also desired. We provide expressions for the slope, incidence angle, and normal vector estimates. Then, we derive the Cram & eacute;r-Rao lower bounds (CRLBs) of these estimates based on error propagation from the CRLBs of the depth estimates. Simulated SAS data supports our results by approaching the theoretical CRLB, and real data examples demonstrate that our results are applicable in practice.

Interval methods have become popular for finding reliable bounds on power patterns for arrays affected by bounded errors. Such bounds can be used in the evaluation and design of antenna and transducer arrays, such as, to determine the worst-case sidelobe level and identify detrimental error configurations. The uncertainty of the element responses can be represented as regions in the complex plane. Previous interval arithmetic methods either wrap these complex intervals into simple shapes before summing or approximate the power pattern bounds directly. The bounds given by these methods are relaxed, which makes the evaluation pessimistic. Tenuti et al. (2017) addressed this by introducing the polygon arithmetic that can approximate bounds with arbitrary tightness at the cost of significantly increased computational complexity. In this work, we show that how the exact bounds can be achieved, and how the complexity can be reduced. We introduce a generalization of the polygon by "inflating" the vertices. This generalization is named polyarc and can exactly represent complex intervals bounded by arcs and edges. Compared to the polygon, polyarc offers simultaneously tighter bounds and reduced complexity; thereby increasing the utility of the interval arithmetic tolerance analysis in the fields of communication, sensing, and imaging.

For years, the research community has viewed adaptive beamforming as a promising approach to enhance image quality in pulse-echo imaging applications such as medical ultrasound and sonar. Methods like the minimum-variance beamformer offer the potential for improved resolution and contrast, enabling sharper representation of key features in the image. However, a significant challenge lies in the high computational cost, stemming from the construction and manipulation of the spatial covariance matrix, which is typically calculated on a per-pixel basis. To address this, several low-complexity methods have been developed that retain the essential benefits of the minimum-variance beamformer while achieving computational efficiency suitable for real-time applications. A fundamental aspect is how to universally apply such beamformers, as the pulse-echo nature allows double adaptiveness over the transmit and/or receive beamforming steps. This presentation provides an overview of the theory and practical applications of these approaches, drawing on over 15 years of research at the University of Oslo and elsewhere. We start with the foundational principles of the minimum-variance beamformer, discuss the motivation for transitioning to low-complexity adaptive beamforming, assess the performance in real-world scenarios, and conclude with recent findings and future directions for this field.

In ultrasound imaging, speckle originates from a large amount of sub-resolution scatterers within the medium. In idealized cases, the speckle envelope statistics follow a Rayleigh distribution, but in practical pulse-echo imaging, the distribution depends on both the imaging system and the underlying tissue structure. Estimating envelope statistics is part of quantitative ultrasound workflows and is also important for image quality assessment as it relates to lesion and tissue detectability. A concrete example is the generalized contrast-to-noise ratio (gCNR), which is a functional of two pixel-value probability density functions (PDFs) from different speckle regions. Such speckle PDFs have, by convention, been estimated from data using histograms, but the accuracy of these estimates can be affected by the nontrivial selection and tuning of the binning parameters. However, the statistics literature widely advocates kernel density estimation (KDE) as a better alternative to histogram-based approaches. In this article, we propose applying a KDE-based method to estimate speckle PDFs in medical ultrasound imaging. The method is practically tuning-free and leverages the Box-Cox transformation to achieve best-in-class performance across a wide range of test cases, and is also robust in cases where gCNR estimation may otherwise fail, such as for skewed distributions that may arise with adaptive beamformers. Furthermore, this work highlights theoretical aspects related to the estimation of PDFs and derived quantities, including the gCNR.

Diverging-wave transmission is a technique used to enhance temporal resolution, but this comes with inherent limitations in terms of image quality. To address these, various specialized and adaptive beamforming methods have been proposed. In the current study, we demonstrate the feasibility of applying the Iterative Adaptive beamforming Approach (IAA) to coherently compounded diverging-wave transmission data - hence mimicking the previously demonstrated focused-transmit setting. This is done on both synthetic simulated and recorded phantom data. The results demonstrate that the proposed method offers comparable contrast and significantly better resolution than the delay-and-sum (DAS) beamformer applied to the same dataset. Moreover, we demonstrate that the IAA method is also applicable to single-transmit diverging waves. Although this setting is quite different from the original IAA regime, the method produces images of similar or better quality than DAS.

Elastography is an imaging technique in which the elastic properties of tissues are estimated and displayed. These are then used in, for example, the diagnosis and staging of various diseases. In the current paper, we describe our contribution to the Algorithms for Mapping Elastic Modulus (a-MEM) Challenge 2024 which is submitted to the category of propagating shear waves from harmonic (continuous) vibration sources. A method that employs directional filters followed by spatial autocorrelation was applied to the data shared by the challenge organizers. In the applied method, directional filters oriented along 12 different angles were employed to decompose the displacement field into shear waves traveling in different directions. A shear-wave speed estimation method based on spatial autocorrelation was applied to filtered complex-valued images. Final shear-wave speed estimates are generated by averaging all shear-wave speed estimates from different directional filter angles and transmission angles.The results showed a good correlation between the given and estimated inclusion coordinates and the method showed a potential that warrants further assessment with in vivo experiments.

Estimating shear wave speed in human tissues provides valuable insights for diagnosing various diseases. These estimates are used, for instance, to determine the stage of certain diseases. The implementation of more accurate and robust data processing methods has the potential to increase the reliability of elastography imaging. In this paper, we propose an algorithm for time-harmonic elastography that utilizes a two-dimensional velocity vector as opposed to using only one dimension. The estimated velocity vectors are projected onto a line that is perpendicular to the shear wave propagation direction. After directional filtering along different angles, the shear wave speed was estimated using a spatial autocorrelation method.We compared the proposed algorithm with the phase-gradient approach using synthetic data generated from k-Wave simulations of a speckle phantom with a stiff inclusion. The results showed that the proposed method has a 49% and 73% lower bias in the inclusion and background regions, respectively. The proposed method shows promising results with lower variance and bias; and further evaluation is underway using in vivo and phantom recording data.

The aim of this study was to adapt a treadmill-developed method for determination of inner-cycle parameters in cross-country roller ski skating for a field application. The method is based on detecting initial and final ground-contact of poles and skis during cyclic movements. Eleven athletes skied four laps of 2.5 km at low and high endurance-intensity, using two types of skis with different rolling coefficients. Participants were equipped with inertial measurement units (IMUs) attached to their wrists and skis, while insoles with pressure sensors and poles with force measurements were used as reference systems. The method based on IMUs was able to detect more than 97% of the temporal events compared to the reference system. The inner-cycle temporal parameters had a precision ranging from 49 to 59 ms, corresponding to 3.9% to 13.7% of the corresponding inner-cycle duration. Overall, this study showed good reliability of using IMUs on athlete’s wrists and skis to determine temporal events, inner-cycle parameters and the performed sub-techniques in cross-country roller ski skating in field-conditions.

Over the past decade, interval arithmetic (IA) has been used to determine tolerance bounds of phased-array beampatterns. IA only requires that the errors of the array elements are bounded and can provide reliable beampattern bounds even when a statistical model is missing. However, previous research has not explored the use of IA to find the error realizations responsible for achieving specific bounds. In this study, the capabilities of IA are extended by introducing the concept of "backtracking," which provides a direct way of addressing how specific bounds can be attained. Backtracking allows for the recovery of the specific error realization and corresponding beampattern, enabling the study and verification of which errors result in the worst-case array performance in terms of the peak sidelobe level (PSLL). Moreover, IA is made applicable to a wider range of arrays by adding support for arbitrary array geometries with directive elements and mutual coupling in addition to element amplitude, phase, and positioning errors. Last, a simple formula for approximate bounds of uniformly bounded errors is derived and numerically verified. This formula gives insights into how array size and apodization cannot reduce the worst-case PSLL beyond a certain limit.

Balance impairment is frequent in people with multiple sclerosis (pwMS) and affects risk of falls and quality of life. By using inertial measurement units (IMUs) on the Single Leg Stance Test (SLS) we aimed to discriminate healthy controls (HC) from pwMS and detect differences in balance endurance and quality. Thirdly, we wanted to test the correlation between instrumented SLS parameters and self-reported measures of gait and balance. Fifty-five pwMS with mild (EDSS<4) and moderate disability (EDSS≥4) and 20 HC performed the SLS with 3 IMUs placed on the feet and sacrum and filled the Twelve Item Multiple Sclerosis Walking Scale (MSWS-12) questionnaire. A linear mixed model was used to compare differences in the automated balance measures. Balance duration was significantly longer in HC compared to pwMS (p < 0.001) and between the two disability groups (p < 0.001). Instrumented measures identified that trunk stability (normalized mediolateral and antero-posterior center of mass stability) had the strongest association with disability (R2 marginal 0.30, p < 0.001) and correlated well with MSWS-12 (R = 0.650, p < 0.001). PwMS tended to overestimate own balance compared to measured balance duration. The use of both self-reported and objective assessments from IMUs can secure the follow-up of balance in pwMS.